The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.

The generating circles of these hypocycloids were later named Cardano circles or cardanic circles and were used for the construction of the first high-speed printing presses.

By the mid 1950s, construction of traffic circles and rotaries had ceased entirely.

Another method of construction uses Carlyle circles, as shown below.

The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.

For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.

Descartes completed ( in 1643 ) the theorem of the three circles of Apollonius, leading to a quadratic equation in 87 terms, each of which is a product of six factors ( which, with this method, makes the actual construction humanly impossible ).

If the radical center lies outside of all three circles, then it is the center of the unique circle ( the radical circle ) that intersects the three given circles orthogonally ; the construction of this orthogonal circle corresponds to Monge's problem.

Later remains include stone circles and cairns of the Late Prehistoric, and even 19th Century remains from the construction of the Southern Pacific, one of the nation's first transcontinental railroads.

The development of new forms of geometric projection in the construction of perspective corresponds with the invention of novel pictorial art forms of visual representation in the Italian Renaissance, since the fourteenth century and up till the end of the sixteenth century, and specifically within the circles of architectural and artistic experimentation and design.

At the beginning is the well-known generalization of Euclid I. 47, then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two.

A widely circulated story holds that the word gadget was " invented " when Gaget, Gauthier & Cie, the company behind the repoussé construction of the Statue of Liberty ( 1886 ), made a small-scale version of the monument and named it after their firm ; however this contradicts the evidence that the word was already used before in nautical circles, and the fact that it did not become popular, at least in the USA, until after World War I.

The final phase of stone circle construction took place in the early to middle Bronze Age ( c. 2200 – 1500 BC ) and saw the construction of numerous small circles which, it has been suggested, were built by individual family groups rather than the large numbers that monuments like Avebury would have required.

or 6 times the area of the circles used in the construction.

Instead, they began the construction of large wooden or stone circles, with many hundreds being built across Britain and Ireland over a period of a thousand years.

The construction is based on two focal circles (" wheels with spokes ")-the western circle is centered on the Cape Verde islands, the eastern focal circle is placed in India.

Start with three circles C < sub > 1 </ sub >, C < sub > 2 </ sub > and C < sub > 3 </ sub >, each one of which is tangent to the other two ( in the general construction, these three circles can be any size, as long as they have common tangents ).

Continuing the construction stage by stage in this way, we can add 2 · 3 < sup > n </ sup > new circles at stage n, giving a total of 3 < sup > n + 1 </ sup > + 2 circles after n stages.

SBA loans, which may be made to small manufacturers, small business pools, wholesalers, retailers, service establishments and other small businesses ( when financing is not otherwise available to them on reasonable terms ), are to finance business construction, conversion, or expansion ; ;

They are conscious of this state's new feeling of optimism and assurance and are definitely impressed by the number of new plants and construction projects in Rhode Island.

-- Approximately 45% of the expenditures for the Department of Defense are for procurement, research, development, and construction programs.

The decreases, which are largely in construction and in aircraft procurement, are offset in part by increases for research and development and for procurement of other military equipment such as tanks, vehicles, guns, and electronic devices.

Industry sources are now estimating that 75% of the signs made during the 1960's will be of plastic construction.

At this time, however, there are signs that increased availability of mortgage credit will not act with the usual speed to stimulate a sharp rise in residential construction.

not like the cracker-box construction of so many of the motel units that have stucco all over the outside but walls that are thin enough so you can hear every movement of the people in the adjoining apartment.

While availability of mortgage money has been a factor in encouraging apartment construction, the generally high level of prosperity in the past few years plus rising consumer income are among the factors that have encouraged builders to concentrate in the apartment-building field.

The structures housing the apartments are of masonry and frame construction.

Architects ' fee structures are typically based on a percentage of construction value, hourly rates or a fixed lump sum fee.

Fixed fees are usually based on a project's allocated construction cost and can range between 4 and 12 % of new construction cost, for commercial and institutional projects, depending on a project's size and complexity.

There are seven districts ( with an eighth currently under construction ), each of which is divided into smaller suburbs, and most of which have a town centre which is the focus of commercial and social activities.

In AmE, collective nouns are almost always singular in construction: the committee was unable to agree.

the inclusion of the term " indivisible " in the Pledge of Allegiance to the United States flag ); before this, the construction " the United States are " was more common.

There are many variations in construction and materials used in steel-string guitars.

All of the guitars above are relatively traditional in looks and construction, and are commonly referred to as " flattop " guitars.

The construction boom has raised many environmental concerns and both the local autonomous government and city council are under scrutiny by the European Union.

New facilities are currently under construction by Fife Council, which will much improve the beach throughout the year.

The plot revolves around a sinister construction corporation's plans to depopulate, destroy and redevelop the Bronx, and a band of rebels who are out to expose the corporation's murderous ways and save their homes.

These investments are concentrated primarily in vehicle assembly and parts supply, the construction industry / cement, raw materials processing / aluminum and regional dairy farming.

The color of the lamps varies by jurisdiction ; typical colors are blue and / or red for police, fire, and medical-emergency vehicles ; amber for tow trucks, security personnel, and construction vehicles ; green for volunteer firefighters, and violet for funerary vehicles.

This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry.

He manages to find some rational points on these curves – elliptic curves, as it happens, in what seems to be their first known occurrence — by means of what amounts to a tangent construction: translated into coordinate geometry

More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle.

Apollonius's genius reaches its highest heights in Book v. Here he treats of normals as minimum and maximum straight lines drawn from given points to the curve ( independently of tangent properties ); discusses how many normals can be drawn from particular points ; finds their feet by construction ; and gives propositions that both determine the center of curvature at any point and lead at once to the Cartesian equation of the evolute of any conic.

In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V ( and more generally ).

The three generating circles, and hence the entire construction, are determined by the location of the three points where they are tangent to one another.

For a curve C given by a sufficiently smooth parametric equations ( twice continuously differentiable ), the osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on C as these points approach P. This is entirely analogous to the construction of the tangent to a curve as a limit of the secant lines through pairs of distinct points on C approaching P.

A simple explicit example of the construction and utility of the-product ( for the simplest case of a two-dimensional euclidean phase space ) is given in the article on Weyl quantization: Two Gaussians compose with this-product according to a hyperbolic tangent law,

* Savage Shapes ( previously Vicious Circles and Other Savage Shapes, ISBN 0-439-99747-X ( signs in geometric diagrams, Loci, constructions: perpendicular bisectors ; dropping perpendiculars ; bisecting angles, triangles: similar ; congruent ; equal areas, polygons: regular ; irregular ; angle sizes and construction, tessellations and Penrose Tiles, origami, circles: chord ; tangent ; angle theorems, regular solids, Euler's formula, ellipses, Geometric proof of Pythagoras ' Theorem.